Step 1

To make the brioche doughnuts: Put the warm milk in a mixing bowl (an electric mixer works best for this recipe--so if you have one put the milk in the mixer bowl). Sprinkle the yeast over the warm milk and allow it to dissolve. Whisk 1 egg and 1 cup of the flour into the yeast mixture. When the dough-sponge is smooth, sprinkle it with an additional cup of flour. Allow the dough-sponge to rise in a warm place until the top layer of flour cracks, about 30 minutes.

Step 2

Lightly beat the 4 remaining eggs. Then, using the dough hook attachment of an electric mixer set at medium speed, or a wooden spoon, work the eggs into the dough. When the dough is smooth, add the sugar, salt, and remaining 1 1/2 cups of flour all at once. If using a mixer, start on low and gradually increase the speed as the dough comes together, mixing for a good 15 to 20 minutes. If you do not have a mixer turn the dough out onto a clean, floured work surface and knead until it is smooth, elastic, and no longer sticky. (Don't be alarmed if the dough seems too wet. It will tighten up into nice, soft, elastic dough.) When the dough comes together, add the butter and mix for another 10 to 15 minutes.

Step 3

Cover the dough with a clean towel and set it aside in a warm place to rise until doubled in size, about 2 hours. Stretch the dough to release some of the trapped gasses and redistribute the yeast, cover the bowl with plastic wrap and refrigerate overnight.

Step 4

Turn the dough out onto a floured work surface. Gently stretch the dough into a rectangle, then roll it out about 3/4-inch thick. Cut the dough with a doughnut cutter. Transfer the doughnuts to a floured board or baking sheet. Cover the doughnuts with a clean towel and allow them to rise in a warm place until they feel soft and fluffy, about 1 hour.

Step 5

To make the crazy glaze: Beat the egg whites until frothy. Add the vanilla and confectioners' sugar and beat until smooth. Divide the glaze between several bowls then color with drops of food coloring.

Step 6

To fry the doughnuts: Heat 2 to 3-inches of oil in a heavy, high-sided pot over medium heat until the oil reaches 350 degrees F. Working in batches of 3 or so, drop the doughnuts into the oil and fry until they float. Turn the doughnuts over in the oil and continue cooking. Cook the doughnuts, turning them once or twice more as necessary, until they are uniformly browned, then transfer them to a plate lined with paper towels to drain.

Step 7

While the doughnuts are still warm, dip 1 side of each into the desired color of glaze then set aside on a wire rack until the glaze is firm. Serve warm or at room temperature.